In their comments, Kathy Laskey and Herman Bruyninckx question
the rationale for the concepts of t-norm and t-conorm. These concepts
are rooted in the work of Karl Menger in the early fifties on probabilistic
metric spaces. For a recent exposition see the treatise "Triangular
Norms," by Klement, Mesiar and Pap, Kluwer, 2000.
It is indeed the case that t-norms and t-conorms are not needed
when events are crisp, in which case standard definitions of
conjunction and disjunction are sufficient. The need arises when events
are imprecise, i.e., fuzzy, as they are in most realistic settings.
Example: What is the conditional probability that A has heart disease if
A has shortness of breath. Note that both "heart disease" and
"shortness of breath" are fuzzy events in the sense that both are
matters of degree. In this case, the conjunction of "heart disease" and
"shortness of breath" can be defined in a variety of ways. More
specifically, the conjunction is a t-norm, that is, a binary connective
satisfying certain natural conditions.
The standard axiomatic structure of standard probability theory
does not address two basic issues which show their ungainly faces in
many real-world applications of probability theory.They are (a)
imprecision of probabilities; and (b) imprecision of events. To address
these issues and to add to probability theory the capability to deal
with perception-based information, e.g., "usually Robert returns from
work at about 6 pm; what is the probability that Robert is home at 6:30
pm?" It is necessary to generalize probability theory in three stages. A
preliminary account of such generalization is described in a forthcoming
paper of mine in the Journal of Statistical Planning and Inference,
"Toward a Perception-based Theory of Probabilistic Reasoning with
Imprecise Probabilities."
-- Professor in the Graduate School, Computer Science Division Department of Electrical Engineering and Computer Sciences University of California Berkeley, CA 94720 -1776 Director, Berkeley Initiative in Soft Computing (BISC)Address: Computer Science Division University of California Berkeley, CA 94720-1776 Tel(office): (510) 642-4959 Fax(office): (510) 642-1712 Tel(home): (510) 526-2569 Fax(home): (510) 526-2433, (510) 526-5181 zadeh@cs.berkeley.edu http://www.cs.berkeley.edu/People/Faculty/Homepages/zadeh.html
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